How do you get Stellarys?

Those are three separate routes, and which one you take depends on what you already have. If you are exploring the Forest of Falling Stars anyway, a wild Stellarys is the direct option. If you have been putting Lumin Amber into a Branch, nurturing can hand you one without the hunt.

The Alpha version is the harder encounter. It also pays more: first clears of Alpha creatures are one of the places Irisalis Petals come from, so the first time you beat Alpha Stellarys is not just a catch.

Branch nurturing has its own gate. Regional nurture opens only once the Branch reaches level 30 and you submit 30 Lumin Amber there, and how much you can nurture each day is capped until you upgrade the branch's nodes. The full system is covered in the linked Branch article.

Why catching Stellarys in the forest counts for more

Capturing the Aniimo found in a region advances that region's capture progression. A Stellarys caught in the Forest of Falling Stars counts toward that area's badges: 40% of the species earns the Bronze Capture Medal and a Vein Essence, and the silver badge needs 70% and also awards Iris petals.

That ties into Prismana. Reaching full gold completion in an area makes Prismana energy build up faster there, so filling the forest's dex, Stellarys included, is how you get the Prismana bonus for that area.

Not settled: The facts do not say whether a Stellarys from Branch nurturing counts toward the Forest of Falling Stars capture progress the way a wild catch does.